Learning Outcomes

On successful completion of this module students should be able to:

1. sketch the graphs of simple functions
2. calculate the limits of real valued functions;
3. determine whether given functions are continuous or not;
4. explain the idea of derivative and compute derivatives from first principles;
5. explain the notion of inverse function;
6. derive the formulae for the derivative of products and quotients of functions;
7. compute the derivative of functions;
8. determine the local maxima and minima of functions and their points of inflexion;
9. compute integrals by the methods of substitution and integration by parts;
10. compute integrals of rational functions and trigonometric functions.

Content

1. FUNCTIONS OF A REAL VARIABLE: Graphs of elementary functions (polynominal, trigomometric, exponential, logarithm, absolute value, integer part). Periodic functions, even and odd functions. Operations on functions: addition, multiplication, division, composition.
2. LIMITS AND CONTINUITY. Limit notation. Rules for manipulation of limits. Sandwich theorem for limits, applications. Definition of continuity at a point in terms of limits. Continuity of sum, product, quotient and composite of continuous functions. Intermediate Value Theorem.
3. DIFFERENTIATION. Fermat’s difference quotient (f(x) –f(a))/(x-a). Definition of the derivative at a point. Geometric significance of the derivative. Differentiation from first principles of some elementary functions. Continuity of a differentiable function; examples of continuous non-differentiable functions. Rules for differentiation. Examples of differentiation, including logarithmic differentiation. Second order derivatives.
4. INVERSE FUNCTIONS. Definition. Trigonometric and polynomial examples. Differentiation of elementary inverse functions.
5. LOCAL MAXIMA AND MINIMA, CURVE SKETCHING. Locating the critical points of a function. Using the first derivative test to determine local maxima and minima. Points of inflexion. Graphs of rational functions, vertical asymptotes, horizontal asymptotes.
6. INTEGRATION. The Fundamental Theorem of Integral Calculus. Linearity of integration. Indefinite integrals. Methods of integration: integration by substitution, integration by parts. Definition of log x as an integral. The exponential function as the inverse of log. The hyperbolic functions. Integral of rational functions, use of partial fractions.